New multivariate cryptosystems are introduced. Sequences f(n) of bijective polynomial transformations of bijective multivariate transformations of affine spaces K^{n}, n = 2, 3, ... , where K is a finite commutative ring with special properties, are used for the constructions of cryptosystems. On axiomatic level, the concept of a family of multivariate maps with invertible decomposition is proposed. Such decomposition is used as private key in a public key infrastructure. Requirements of polynomiality of degree and density allow to estimate the complexity of encryption procedure for a public user. The concepts of stable family and family of increasing order are motivated by studies of discrete logarithm problem in Cremona group. Statement on the existence of families of multivariate maps of polynomial degree and polynomial density with the invertible decomposition is formulated. We observe known explicit constructions of special families of multivariate maps. They correspond to explicit constructions of families of nonlinear algebraic graphs of increasing girth which appeared in Extremal Graph Theory. The families are generated by pseudorandom walks on graphs. This fact ensures the existence of invertible decomposition; a certain girth property guarantees the increase of order for the family of multivariate maps, good expansion properties of families of graphs lead to good mixing properties of graph based private key algorithms. We describe the general schemes of cryptographic applications of such families (public key infrastructure, symbolic Diffie—Hellman protocol, functional versions of El Gamal algorithm).
Ding J. , Gower, J. E. and Schmidt, D. S., Multivariate Public Key Cryptosystems, 260. Springer, Advances in Information Security, v. 25, (2006).
Bollobás, B., Extremal Graph Theory, Academic Press, London, 1978.
Erdős, P., Rényi, A. and T. Sós, V., On a problem of graph theory, Studia. Sci. Math. Hungar., 1 (1966), 215–235.
Erdős, P. and Simonovits, M., Compactness results in extremal graph theory, Combinatorica, 2 (3), 275–288 (1982).
Simonovits, M., Extermal Graph Theory, in: Selected Topics in Graph Theory, 2, edited by L. W. Beineke and R. J. Wilson, Academic Press, London, pp. 161–200 (1983).
Ustimenko, V., Coordinatisation of Trees and their Quotients, in: Voronoj’s Impact on Modern Science, Kiev, Institute of Mathematics, vol. 2, 125–152 (1998).
Ustimenko, V., CRYPTIM: Graphs as Tools for Symmetric Encryption, Lecture Notes in Computer Science, Springer, v. 2227, 278–287 (2001).
Ustimenko, V., Maximality of affine group and hidden graph cryptosystems, J. Algebra Discrete Math., No 1, P. 133–150 (2005).
Wróblewska, A., On some properties of graph based public keys, Albanian Journal of Mathematics, Volume 2, Number 3, 2008, 229–234, NATO AdvancedStudies Institute: “New challenges in digital communications”.
Ustimenko, Vasyl and Wróblewska, Aneta, On some algebraic aspects of data security in cloud computing, Proceedings of International conference Applications of Computer Algebra, Malaga, p. 144–147 (2013).
Romańczuk, U. and Ustimenko, V., On regular forests given in terms of algebraic geometry, new families of expanding graphs with large girth and new multivariate cryptographical algorithms, Proceedings of International conference Applications of Computer Algebra, Malaga, p. 135–139 (2013).
Margulis, G., Explicit group-theoretical constructions of combinatorial schemes and their application to desighn of expanders and concentrators, Probl.Peredachi Informatsii., 24, N1, 51–60. English translation publ. Journal ofProblems of Information transmission, 39–46 (1988).
Lubotsky, A., Philips, R. and Sarnak, P., Ramanujan graphs, J. Comb. Theory, 115, N 2 (1989), 62–89.
Lazebnik, F., Ustimenko, V. A. and Woldar, A. J., A New Series of Dense Graphs of High Girth, Bull (New Series) of AMS, v. 32, N1 (1995), 73–79.
Guinand, P. and Lodge, J., Tanner type codes arising from large girth graphs, Canadian Workshop on Information Theory CWIT ’97, Toronto, Ontario, Canada, 5–7 (June 3–6 1997).
MacKay, D. and Postol, M., Weakness of Margulis and Ramanujan–Margulis Low Dencity Parity Check Codes, Electronic Notes in Theoretical ComputerScience, 74 (2003), 8 pp.
Ustimenko, V., On some optimisation problems for graphs and multivariate cryptography (in Russian), in: Topics in Graph Theory: A tribute to A. A. and T. E. Zykova on the ocassion of A. A. Zykov birthday, pp. 15–25, 2013, www.math.uiuc.edu/kostochka.
Ustimenko, V. A., On extremal graph theory and symbolic computations, Dopovidi National Academy of Sci of Ukraine, N2 (2013), 42–49 (in Russian).
Polak, M. and Ustimenko, V. A., On LDPC Codes Corresponding to Infinite Family of Graphs A(n,K), Proceedings of the Federated Conference on Computer Science and Information Systems (FedCSIS), CANA, Wroclaw, pp. 11–23 (September, 2012).
Ustimenko, V., On the extremal graph theory for directed graphs and its cryptographical applications, in: T. Shaska, W. C. Huffman, D. Joener and V. Ustimenko, Advances in Coding Theory and Cryptography, Series on Coding andCryptology, vol. 3, 181–200 (2007).
Kotorowicz, J. and Ustimenko, V., On the implementation of cryptoalgorithms based on algebraic graphs over some commutative rings, Condenced Matters Physics, Special Issue: Proceedings of the international conferences “Infiniteparticle systems, Complex systems theory and its application”, KazimerzDolny, Poland, 2006, 11 (no. 2(54)) 347–360 (2008).
Ustimenko, V. A. and Romańczuk, U., On Extremal Graph Theory, Explicit Algebraic Constructions of Extremal Graphs and Corresponding Turing Encryption Machines, in: Artificial Intelligence, Evolutionary Computing andMetaheuristics, In the footsteps of Alan Turing Series: Studies in Computational Intelligence, Vol. 427, Springer, 257–285 (January, 2013).
Ustimenko, V. A. and Romańczuk, U., On Dynamical Systems of Large Girth or Cycle Indicator and their applications to Multivariate Cryptography, in: ArtificialIntelligence, Evolutionary Computing and Metaheuristics, In the footsteps of Alan Turing Series: Studies in Computational Intelligence, Volume 427, 257–285 (January 2013).
Klisowski, M. and Ustimenko, V. A., On the Comparison of Cryptographical Properties of Two Different Families of Graphs with Large Cycle Indicator, Mathematics in Computer Science, Volume 6, Number 2, Pages 181–198(2012).
Ustimenko, V. A., On the cryptographical properties of extreme algebraic graphs, in: Algebraic Aspects of Digital Communications, IOS Press (Lectures of Advanced NATO Institute, NATO Science for Peace and Security Series-D: Information and Communication Security, Volume 24, 296 pp (July 2009).
Ustimenko, V., On Multivariate Cryptosystems Based on Computable Maps with Invertible Decompositions, Annales of UMCS, Informatica, volume 14, Specialissue Proceedings of International Conference Cryptography and Security Systems, pp. 7–18 (2014).
Ustimenko, V. A., On multivariate cryptosystem based on maps with logarithmically invertible decomposition corresponding to walk on graph, FedSCIS 2014 Proceedings (to appear as regular paper).
Ustimenko, V. A. and Wróblewska, A., On new examples of families of multivariate maps and their cryptographical applications, Proceedings of the Conferenceon Security Systems, Lublin, Annales of UMCS (to appear) (2014).
Romańczuk-Polubec, U. and Ustimenko, V. A., On multivariate maps based on polynomially compressed maps with invertible decomposition, Proceedings of the Conference on Security Systems, 2014, Lublin, in: Cryptography and SecuritySystems, Third International Conference, CSS 2014, Lublin, Poland, September 22–24 (2014). Proceedings, Communications in Computer and Information Science, 448, p. 23–37.
Polak, M. and Ustimenko, V., On LDPC codes based on families of expanding graphs of increasing girth without edge transitive automorphism Groups, Proceedings of the Third International Conference CSS 2014, Lublin, Poland, September 22–24, 2014, Communications in Computer and Information Science, 448, p. 23–37.
Morgenstern, M., Existence and explicit constructions of q + 1-regular Ramanujan graphs for every prime power q, Journal of Combinatorial Theory, Ser. B, (62), no 1 (1994), 44–62.
Lazebnik, F., Ustimenko, V. A. and Woldar, A. J., Polarities and 2k-cycle-free graphs, Discrete Mathematics, 197/198 (1999), 503–513.
Romańczuk-Polubiec, U. and Ustimenko, V., On two windows multivariatecryptosystem depending on random parameters, Algebra and Discrete Mathematics, volume 19, N1, pp. 101–129 (2015).
Klisowski, M., Improvement of the security of cryptographic multivariate algorithms based on algebraic graph theory, PhD Thesis, Chenstohowa, 112 pp(March, 2014).